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\begin{document}

\title{Complex Analysis}
\subtitle{Chapter 2. Complex Functions \\
Section 3. The Exponential and Trigonometric Functions}
%\institute{SLUC}
\author{LVA}
%\date
%\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{ {2023年9月21日} }

\maketitle

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% 设置方程编号从16开始
\setcounter{equation}{19} % 因为设置为19，下一个使用的标签就会从20开始

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\begin{frame}{Contents 1-2} 

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item  Introduction to the Concept of Analytic Function
\begin{enumerate}
\item[1.1.] Limits and Continuity
\item[1.2.] Analytic Functions
\item[1.3.] Polynomials
\item[1.4.] Rational Functions
\end{enumerate}

\item Elementary Theory of Power Series
\begin{enumerate}
\item[2.1.] Sequences
\item[2.2.] Series
\item[2.3.] Uniform Convergence
\item[2.4.] Power Series
\item[2.5.] Abel's Limit Theorem
\end{enumerate}

\end{enumerate}

\end{frame}

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\begin{frame}{Contents 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}

\item[3.] {\color{red}The Exponential and Trigonometric Functions}
\begin{enumerate}
\item[3.1.] {\color{red}The Exponential }
\item[3.2.] {\color{red}The Trigonometric Functions}
\item[3.3.] {\color{red}The Periodicity}
\item[3.4.] {\color{red}The Logarithm}
\end{enumerate}

\end{itemize}

\end{frame}


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\begin{frame}{3. The Exponential and Trigonometric Functions }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[1.] 
The person who approaches calculus exclusively from the point of view of real numbers will not expect any relationship between the exponential function $e^x$ and the trigonometric functions $\cos x$ and $\sin x$. 

\item[2.] 
Indeed, these functions seem to be derived from completely different sources and with different purposes in mind. 

\item[3.] 
{\color{red}He will notice, no doubt, a similarity between the Taylor developments of these functions, and if willing to use imaginary arguments he will be able to derive {\color{blue} Euler's formula} $e^{ix} = \cos x + i \sin x$ as a formal identity. }

\item[4.] 
But it took the genius of a Gauss to analyze its full depth. 

\end{enumerate}

\end{frame}

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\begin{frame}{3. The Exponential and Trigonometric Functions }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[5.] 
With the preparation given in the preceding section it will be easy to define $e^z$, $\cos z$ and $\sin z$ for complex $z$, and to derive the relations between these functions. 

\item[6.] 
At the same time we can define the logarithm as the inverse function of the exponential, and the logarithm leads in turn to {\color{blue}the correct definition of the argument} of a complex number, and hence to {\color{blue}the nongeometric definition of angle}.

\end{enumerate}

\end{frame}

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\begin{frame}{3.1. The Exponential. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[1.] 
{\color{red}We may begin by defining the {\color{blue}exponential function} as the solution of the differential equation
\begin{equation}
f'(z) = f(z)
\label{eq-20}
\end{equation}
with the initial value $f(0)=1$. 
}
 
\item[2.] 
We solve it by setting
\begin{equation*}
\begin{aligned}
f(z) &= a_0 + a_1z + \cdots + a_nz^n + \cdots \\ 
f'(z) &= a_1 + 2a_2z + \cdots + na_nz^{n-1} + \cdots .
\end{aligned}
\end{equation*}

\item[3.] 
If (\ref{eq-20}) is to be satisfied, we must have $a_{n-1} = na_n$, and the initial condition gives $a_0 = 1$. 

\item[4.] 
It follows by induction that $a_n = 1/n!$.


\end{enumerate}

\end{frame}

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\begin{frame}{3.1. The Exponential. \hfill 作业2C-1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[5.] 
The solution is denoted by $e^z$ or $\exp z$, depending on purely typographical considerations.

\item[6.] 
We must show of course that the series
\begin{equation}
e^z = 1 + \frac{z}{1!} + \frac{z^2}{2!} + \cdots + \frac{z^n}{n!} + \cdots 
\label{eq-21}
\end{equation}
converges. 

\item[7.] 
{\color{red}It does so in the whole plane, for $\sqrt[n]{n!}\to\infty$.}
%(proof by the reader). 

\end{enumerate}

\end{frame}

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\begin{frame}{3.1. The Exponential. \hfill 作业2C-2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[8.] 
{\color{red}It is a consequence of the differential equation that $e^z$ satisfies the {\color{blue} addition theorem}
\begin{equation}
e^{a+b} = e^a\cdot e^b. 
\label{eq-22}
\end{equation}
}

\item[9.] 
Indeed, we find that $D(e^z\cdot e^{c-z}) = e^z\cdot e^{c-z} + e^z\cdot (-e^{c-z})=0$. 

\item[10.] 
Hence $e^z\cdot e^{c-z}$ is a constant. 

\item[11.] 
The value of the constant is found by setting $z = 0$.

\item[12.] 
We conclude that $e^z\cdot e^{c-z} = e^c$, and (\ref{eq-22}) follows for $z = a$, $c = a+b$.

\end{enumerate}

\end{frame}

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\begin{frame}{3.1. The Exponential. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[13.] 
Remark. We have used the fact that $f(z)$ is constant if $f'(z)$ is identically zero. 

\item[14.] 
This is certainly so if $f$ is defined in the whole plane. 

\item[15.] 
For if $f = u + iv$ we obtain
\begin{equation*}
\frac{\partial u}{\partial x} = \frac{\partial u}{\partial y}
=\frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} = 0, 
%\label{eq-}
\end{equation*}
and the real version of the theorem shows that $f$ is constant on every horizontal and every vertical line. 

\end{enumerate}

\end{frame}

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\begin{frame}{3.1. The Exponential. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[16.] 
As a particular case of the addition theorem $e^z\cdot e^{-z} = 1$.

\item[17.] 
This shows that $e^z$ is never zero. 

\item[18.] 
For real $x$ the series development (\ref{eq-21}) shows that $e^x > 1$ for $x > 0$, and since $e^x$ and $e^{-x}$ are reciprocals, $0 < e^x < 1$ for $x < 0$.

\item[19.] 
The fact that the series has real coefficients shows that $\exp \bar{z}$ is the complex conjugate of $\exp z$. 

\item[20.] 
Hence $|e^{iy}|^2 = e^{iy}\cdot e^{-iy} = 1$, and $|e^{x+iy}| = e^x$.


\end{enumerate}

\end{frame}

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\begin{frame}{3.2. The Trigonometric Functions. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[1.] 
{\color{red}The trigonometric functions are defined by
\begin{equation}
\cos z = \frac{e^{iz}+e^{-iz}}{2}, \hspace{0.5cm}
\sin z = \frac{e^{iz}-e^{-iz}}{2i}
\label{eq-23}
\end{equation}
}

\item[2.] 
Substitution in (\ref{eq-21}) shows that they have the series developments 
\begin{equation*}
\begin{aligned}
\cos z &= 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \cdots   \\ 
\sin z &= z - \frac{z^3}{3!} + \frac{z^5}{5!} - \cdots   \\ 
\end{aligned}
%\label{eq-}
\end{equation*}

\end{enumerate}

\end{frame}

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\begin{frame}{3.2. The Trigonometric Functions. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[3.] 
For real $z$ they reduce to the familiar Taylor developments of $\cos x$ and $\sin x$, {\color{red}with the significant difference that we have now redefined these functions without use of geometry. } 

\item[4.] 
From (\ref{eq-23}) we obtain further {\color{blue} Euler's formula }
\begin{equation*}
e^{iz} = \cos z + i \sin z
\end{equation*}
as well as the identity
\begin{equation*}
\cos^2 z + \sin^2 z = 1.
\end{equation*}

\item[5.] 
It follows likewise that
\begin{equation*}
D \cos z = - \sin z, \hspace{0.5cm}
D \sin z = \cos z.
\end{equation*}

\end{enumerate}

\end{frame}

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\begin{frame}{3.2. The Trigonometric Functions. \hfill 作业2C-3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}


\item[6.] 
{\color{red}
The addition formulas
\begin{equation*}
\begin{aligned}
\cos (a+b) &= \cos a\, \cos b - \sin a\, \sin b \\ 
\sin (a+b) &= \cos a\, \sin b + \sin a\, \cos b
\end{aligned}
\end{equation*}
are direct consequences of (\ref{eq-23}) and the addition theorem for the exponential function.
}

\end{enumerate}

\end{frame}

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\begin{frame}{3.2. The Trigonometric Functions. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[7.] 
The other trigonometric functions $\tan z$, $\cot z$, $\sec z$, $\mathrm{cosec}\, z$ are of secondary importance. 

\item[8.] 
They are defined in terms of $\cos z$ and $\sin z$ in the customary manner. 

\item[9.] 
We find for instance
\begin{equation*}
\tan z = -i \frac{e^{iz} - e^{-iz}}{e^{iz} + e^{-iz}}. 
\end{equation*}

\item[10.] 
{\color{blue}Observe that all the trigonometric functions are rational functions of $e^{iz}$. } 


\end{enumerate}

\end{frame}

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\begin{frame}{3.2. The Trigonometric Functions. Exercise - 1 \hfill 作业2C-4 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Find the values of $\sin i$, $\cos i$, $\tan (1 + i)$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.2. The Trigonometric Functions. Exercise - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The hyperbolic cosine and sine are defined by 
$$
\cosh z = \frac{e^z+e^{-z}}{2},\hspace{0.5cm}
\sinh z = \frac{e^z-e^{-z}}{2}.
$$
Express them through $\cos iz$, $\sin iz$. 
Derive the addition formulas, and formulas for $\cosh 2z$, $\sinh 2z$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.2. The Trigonometric Functions. Exercise - 3 \hfill 作业2C-5 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Use the addition formulas to separate $\cos (x + iy)$, $\sin (x + iy)$ in real and imaginary parts.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.2. The Trigonometric Functions. Exercise - 4}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that
$$
|\cos z|^2 = \sinh^2 y + \cos^2 x = \cosh^2 y - \sin^2 x = \frac{1}{2} (\cosh 2y + \cos 2x)
$$
and
$$
|\sin z|^2 = \sinh^2 y + \sin^2 x = \cosh^2 y - \cos^2 x = \frac{1}{2} (\cosh 2y - \cos 2x). 
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.3. The Periodicity. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[1.] 
{\color{red}We say that $f(z)$ has the period $c$ if $f(z + c) = f(z)$ for all $z$.}

\item[2.] 
Thus a period of $e^z$ satisfies $e^{z+c} = e^z$, or $e^c = 1$. 

\item[3.] 
It follows that $c = i\omega$ with real $\omega$; we prefer to say that $\omega$ is a period of $e^{iz}$.

\item[4.] 
We shall show that there are periods, and that they are all integral multiples of a
positive period $\omega_0$.

\end{enumerate}

\end{frame}

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\begin{frame}{3.3. The Periodicity. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[5.] 
{\color{red}
Of the many ways to prove the existence of a period we choose the following.} 

\item[6.] 
From $D \sin y = \cos y \le 1$ and $\sin 0 = 0$ we obtain $\sin y < y$ for $y > 0$, either by integration or by use of the mean-value theorem. 

\item[7.] 
In the same way $D \cos y = - \sin y > -y$ and $\cos 0 = 1$ gives $\cos y > 1 - y^2/2$, which in turn leads to $\sin y > y - y^3/6$ and finally to $\cos y < 1 - y^2/2 + y^4/24$.

\item[8.] 
This inequality shows that $\cos \sqrt{3} < 0$, and therefore there is a $y_0$ between $0$ and $\sqrt{3}$ with $\cos y_0 = 0$. 


\end{enumerate}

\end{frame}

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\begin{frame}{3.3. The Periodicity. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[9.] 
Because
$$\cos^2 y_0 + \sin^2 y_0 = 1$$
we have $\sin y_0 = \pm 1$, that is, $e^{iy_0} = \pm i$, and hence $e^{4iy_0} = 1$.

\item[10.] 
We have shown that $4y_0$ is a period. 

\item[11.] 
{\color{red}
Actually, it is the smallest positive period.
}

\item[12.] 
To see this, take $0 < y < y_0$. 

\item[13.] 
Then $\sin y > y(1 - y^2/6) > y/2 > 0$, which shows that $\cos y$ is strictly decreasing. 

\end{enumerate}

\end{frame}

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\begin{frame}{3.3. The Periodicity. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}


\item[14.] 
Because $\sin y$ is positive and $\cos^2 y + \sin^2 y = 1$ it follows that $\sin y$ is strictly increasing, and hence $\sin y < \sin y_0 = 1$. 

\item[15.] 
The double inequality $0 < \sin y < 1$ guarantees that $e^{iy}$ is neither $\pm 1$ nor $\pm i$. 

\item[16.] 
Therefore $e^{4iy} \neq 1$, and $4y_0$ is indeed the smallest positive period. 

\item[17.] 
We denote it by $\omega_0$. 

\item[18.] 
Consider now an arbitrary period $\omega$. 

\item[19.] 
There exists an integer $n$ such that $n\omega_0 \le \omega < (n + 1)\omega_0$. 

\end{enumerate}

\end{frame}

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\begin{frame}{3.3. The Periodicity. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[20.] 
If $\omega$ were not equal to $n\omega_0$, then $\omega - n\omega_0$ would be a positive period $< \omega_0$. 

\item[21.] 
Since this is not possible, every period must be an integral multiple of $\omega_0$.

\item[22.] 
{\color{red}
The smallest positive period of $e^{iz}$ is denoted by $2\pi$. 
}

\item[23.] 
In the course of the proof we have shown that
\begin{equation*}
e^{\pi i/2} = i, \,\, 
e^{\pi i} = -1, \,\, 
e^{2\pi i} = 1.  
\end{equation*}

\item[24.] 
{\color{blue}
These equations demonstrate the intimate relationship between the numbers $e$ and  $\pi$.
}

\item[25.] 
When $y$ increases from $0$ to $2\pi$, the point $w = e^{iy}$ describes the unit circle $|w| = 1$ in the positive sense, namely from $1$ over $i$ to $-1$ and back  over $-i$ to $1$.

\end{enumerate}

\end{frame}

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\begin{frame}{3.3. The Periodicity. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[26.] 
For every $w$ with $|w| = 1$ there is one and only one $y$ from the half-open interval $0 \le y < 2\pi$ such that $w = e^{iy}$. 

\item[27.] 
All this follows readily from the established fact that $\cos y$ is strictly decreasing in the ``first quadrant'', that is, between $0$ and $\pi/2$. 

\item[28.] 
From an algebraic point of view the mapping $w = e^{iy}$ establishes a homomorphism between the additive group of real numbers and the multiplicative group of complex numbers with absolute value 1. 

\item[29.] 
The kernel of the homomorphism is the subgroup formed by all integral multiples $2\pi n$. 

\end{enumerate}

\end{frame}

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\begin{frame}{3.4. The Logarithm. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[1.] 
Together with the exponential function we must also study its inverse function, the logarithm. 

\item[2.] 
{\color{red}By definition, $z = \log w$ is a root of the equation $e^z = w$. } 

\item[3.] 
First of all, since $e^z$ is always $\neq 0$, the number 0 has no logarithm. 

\item[4.] 
For $w \neq 0$ the equation $e^{x+iy} = w$ is equivalent to
\begin{equation}
e^x = |w|, \hspace{0.5cm} 
e^{iy} = w/|w|. 
\label{eq-24}
\end{equation}

\item[5.] 
The first equation has a unique solution $x = \log |w|$, the {\color{red}real logarithm} of the positive number $|w|$.

\end{enumerate}

\end{frame}

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\begin{frame}{3.4. The Logarithm. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[6.] 
The right-hand member of the second equation (\ref{eq-24}) is a complex number of absolute value 1.

\item[7.] 
{\color{blue}Therefore, as we have just seen, it has one and only one solution in the interval $0 \le y < 2\pi$. 
}

\item[8.] 
In addition, it is also satisfied by all y that differ from this solution by an integral multiple of $2\pi$.

\item[9.] 
{\color{red}We see that every complex number other than 0 has infinitely many logarithms which differ from each other by multiples of $2\pi i$. }

\end{enumerate}

\end{frame}

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\begin{frame}{3.4. The Logarithm. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[10.] 
{\color{red}The imaginary part of $\log w$ is also called the {\color{blue}argument} of $w$, $\mathrm{arg}\,w$, and it is interpreted geometrically as the angle, measured in radians, between the positive real axis and the half line from 0 through the point $w$. } 

\item[11.] 
According to this definition the argument has infinitely many values which differ
by multiples of $2\pi$, and 
\begin{equation*}
\log w = \log |w| + i\, \mathrm{arg}\, w.
\end{equation*}

\item[12.] 
With a change of notation, if $|z| = r$ and $\mathrm{arg}\, z = \theta$, then $z = re^{i\theta}$. 

\item[13.] 
This notation is so convenient that it is used constantly, even when the exponential function is not otherwise involved.


\end{enumerate}

\end{frame}

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\begin{frame}{3.4. The Logarithm. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[14.] 
{\color{red}
By convention the logarithm of a positive number shall always mean the real logarithm, unless the contrary is stated. 
}

\item[15.] 
The symbol $a^b$, where $a$ and $b$ are arbitrary complex numbers except for the condition $a \neq 0$, is always interpreted as an equivalent of $\exp (b \log a)$. 

\item[16.] 
{\color{orange} 
If $a$ is restricted to positive numbers, $\log a$ shall be real, and $a^b$ has a single value. 
}

\item[17.] 
Otherwise $\log a$ is the complex logarithm, and $a^b$ has in general infinitely many values which differ by factors $e^{2\pi inb}$.

\item[18.] 
There will be a single value if and only if $b$ is an integer $n$, and then $a^b$ can be interpreted as a power of $a$ or $a^{-1}$. 

\end{enumerate}

\end{frame}

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\begin{frame}{3.4. The Logarithm. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[19.] 
If $b$ is a rational number with the reduced form $p/q$, then $a^b$ has exactly $q$ values and can be represented as $\sqrt[q]{a^p}$.

\item[20.] 
The addition theorem of the exponential function clearly implies 
\begin{equation*}
\begin{aligned}
\log (z_1z_2) &= \log z_1 + \log z_2 \\ 
\mathrm{arg}\, (z_1z_2) &= \mathrm{arg}\, z_1 + \mathrm{arg}\, z_2,
\end{aligned}
\end{equation*}
{\color{orange}
but only in the sense that both sides represent the same infinite set of complex numbers. 
}

\item[21.] 
If we want to compare a value on the left with a value on the right, then we can merely assert that they differ by a multiple of $2\pi i$ (or $2\pi$). 

\item[22.] 
(Compare with the remarks in Chap. 1, Sec. 2.1.)


\end{enumerate}

\end{frame}

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\begin{frame}{3.4. The Logarithm. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[23.]  
Finally we discuss the {\color{blue}inverse cosine} which is obtained by solving the equation
\begin{equation*}
\cos z = \frac{1}{2} (e^{iz} + e^{-iz}) = w.
\end{equation*}

\item[24.] 
This is a quadratic equation in $e^{iz}$ with the roots 
\begin{equation*}
e^{iz} = w \pm \sqrt{w^2 - 1}.
\end{equation*}
and consequently
\begin{equation*}
z = \arccos w = - i\log (w \pm \sqrt{w^2-1}).  
\end{equation*}

\end{enumerate}

\end{frame}

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\begin{frame}{3.4. The Logarithm. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}


\item[25.] 
We can also write these values in the form
\begin{equation*}
\arccos w = \pm i \log (w + \sqrt{w^2-1}),
\end{equation*}
for $w+\sqrt{w^2-1}$ and $w-\sqrt{w^2-1}$ are {\color{blue}reciprocal numbers}. 

\item[26.] 
The infinitely many values of $\arccos w$ reflect the {\color{blue}evenness and periodicity} of $\cos z$. 

\item[27.] 
The inverse sine is most easily defined by 
\begin{equation*}
\arcsin w = \frac{\pi}{2} - \arccos w.
\end{equation*}

\end{enumerate}

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\item[28.] 
It is worth emphasizing that in the theory of complex analytic functions all elementary transcendental functions can thus be expressed through $e^z$ and its inverse $\log z$. 

\item[29.] 
In other words, there is essentially only one elementary transcendental function.


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\begin{itemize}
\item  {\color{red}Question. 
For real $y$, show that every remainder in the series for $\cos y$ and $\sin y$ has the same sign as the leading term (this generalizes the inequalities used in the periodicity proof, Sec. 3.3).
}

\item  Answer. 
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\item 
 
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\end{enumerate}

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\begin{itemize}
\item  {\color{red}Question. 
Prove, for instance, that $3 < \pi < 2\sqrt{3}$.
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\item  Answer. 
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\item 
 
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\item 

\end{enumerate}

\end{itemize}

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\begin{itemize}
\item  {\color{red}Question. 
Find the value of $e^z$ for $z=-\frac{\pi i}{2}, \frac{3\pi i}{4}, \frac{2\pi i}{3}$. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
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\item 

\end{enumerate}

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\begin{itemize}
\item  {\color{red}Question. 
For what values of $z$ is $e^z$ equal to $2$, $-1$, $i$, $-i/2$, $-1-i$, $1+2i$?
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\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

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\begin{itemize}
\item  {\color{red}Question. 
Find the real and imaginary parts of $\exp(e^z)$.
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\item  Answer. 
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\item 
 
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\end{enumerate}

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\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Determine all values of $2^i$, $i^i$, $(-1)^{2i}$.
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\item  Answer. 
\begin{enumerate}
\item 
 
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\item 

\end{enumerate}

\end{itemize}

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\begin{frame}{3.4. The Logarithm. Exercise - 7}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Determine the real and imaginary parts of $z^z$.
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\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

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\begin{frame}{3.4. The Logarithm. Exercise - 8}

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\begin{itemize}
\item  {\color{red}Question. 
Express $\arctan w$ in terms of the logarithm.
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\item  Answer. 
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\item 
 
\item 

\item 

\end{enumerate}

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\begin{frame}{3.4. The Logarithm. Exercise - 9 \hfill 作业2C-10 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show how to define the ``angles'' in a triangle, bearing in mind that they should lie between 0 and $\pi$. With this definition, prove that the sum 
of the angles is $\pi$.
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\item  Answer. 
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\item 
 
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\item 

\end{enumerate}

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\begin{itemize}
\item  {\color{red}Question. 
Show that the roots of the binomial equation $z^n = a$ are the vertices of a regular polygon (equal sides and angles).
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\item  Answer. 
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\item 
 
\item 

\item 

\end{enumerate}

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